Large-time behavior for a fully nonlocal heat equation

TL;DR

This paper investigates the long-term behavior of solutions to a nonlocal heat equation involving fractional Laplacian and Caputo time derivative, revealing new results in fast space-time scales due to the equation's fundamental solution tails.

Contribution

It extends previous work by analyzing the large-time behavior of nonlocal heat equations with fractional derivatives, especially in fast scales, providing novel insights not available for classical cases.

Findings

Characterization of large-time behavior in all L^p norms

Analysis of solutions in various space-time scales

Identification of unique behaviors in fast scales due to fat tails

Abstract

We study the large-time behavior in all $L^p$ norms and in different space-time scales of solutions to a nonlocal heat equation in $\mathbb{R}^N$ involving a Caputo $\alpha$-time derivative and a power of the Laplacian $(-\Delta)^s$, $s\in (0,1)$, extending recent results by the authors for the case $s=1$. The initial data are assumed to be integrable, and, when required, to be also in $L^p$. The main novelty with respect to the case $s=1$ comes from the behaviour in fast scales, for which, thanks to the fat tails of the fundamental solution of the equation, we are able to give results that are not available neither for the case $s=1$ nor, to our knowledge, for the standard heat equation, $s=1$, $\alpha=1$.